Summary: We interpolate a new family of Teichmüller spaces \(T_{\sharp}^X\) between the universal Teichmüller space \(T\) and its little subspace \(T_0\). Each \(T_{\sharp}^X\) is defined by prescribing a subset \(X\) of the unit circle as the exceptional set of the vanishing property for \(T_0\). The inclusion relation of \(X\) induces a natural inclusion of \(T_{\sharp}^X\), and an approximation of \(T\) by an increasing sequence of \(T_{\sharp}^X\) is investigated. In this paper, we discuss the fundamental properties of \(T_{\sharp}^X\) from the viewpoint of the quasiconformal theory of Teichmüller spaces. We also consider the quotient space of \(T\) by \(T_{\sharp}^X\) as an analog of the asymptotic Teichmüller space.